Stability of continuation and obstacle problems in acoustic and electromagnetic scattering

Thesis (Ph.D.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics and StatisticsStudy of the Cauchy problem for Helmholtz equation is motivated by the inverse scattering theory and more generally by remote sensing. In this dissertation the increased stability of the Cauchy problem for Helmholtz equation and the Maxwell's system is investigated with varying frequency. Here it has been shown that the the stability of continuation is improving with the increasing frequency. The continuation is inside the convex hull of the surface where the Cauchy data is given. This has been demonstrated by numerical experiments with simple geometry. When we continue outside of the convex hull, the subspace of stable solutions is growing with frequency. This is also demonstrated by numerical experiments where we reconstruct the density function of the single layer potential. Another problem that is presented here is the electromagnetic obstacle scattering problem, with variable frequency. Here the existence and uniqueness of the solution to the forward problem is presented and the analytic dependence of the solution on the frequency is proved